Faculté des sciences

## Plane waves, matrix models and space-time singularities

### Thèse de doctorat : Université de Neuchâtel, 2009 ; Th. 2070.

As the outcome of four years of learning and research in the string theory group at Neuchâtel, this work does not follow a straight line from beginning to end. It is rather to be understood as a melange of several concepts and techniques which could have been arranged in different order and with different emphasis. Consequently, the reader should not feel forced to follow the chosen path, but is... Plus

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Summary
As the outcome of four years of learning and research in the string theory group at Neuchâtel, this work does not follow a straight line from beginning to end. It is rather to be understood as a melange of several concepts and techniques which could have been arranged in different order and with different emphasis. Consequently, the reader should not feel forced to follow the chosen path, but is encouraged to pick out his own topics of interest. Plane waves are one of the major tools employed, introduced in chapter one as the result of the Penrose limit of any space-time as well as interesting gravitational backgrounds themselves. Curved, yet plain enough to allow for many detailed calculations they can be considered the next logical step after flat space. We briefly review many of their attractive features, geometry, symmetry and relevance to light-cone quantisation, stressing the use of Brinkmann coordinates. These are Fermi coordinates on plane waves, and in the publication reprinted at the end of the chapter this notion is exploited to construct the geometric expansion of a space-time about the respective plane wave. Chapter two connects to the first chapter in reviewing the application of the plane wave limit to power-law space-time singularities of the Szekeres-Iyer class, encompassing a wide range of well-known physical solutions. The result is universal: singular homogeneous plane waves, provided that the dominant energy condition (DEC) holds. The following publication builds up on this using functional analytic methods to characterise scalar field probes on the same backgrounds. The criterion of a unique time-evolution classifies singular behaviour of the fields. Departing from the notion of classical space-time, chapter three introduces a new concept: membrane quantisation, a technique to regularise a U(infinity) diffeomorphism subgroup to U(N) matrix theory. We present the basic construction from a new angle detailing gauge-fixing procedure and origin of the gauge field. We also mention the extension to the supersymmetric BMN model and the connection to gauge theory compactification. The fuzzy sphere ground states of the model allow for a fluctuation expansion with complete control over the spectrum. We advocate the use of t'Hooft's R-xi-gauges and explain the implication of their unphysical gauge parameter. The BMN matrix model is embedded into the larger context of string theory dualities and M-theory in chapter four. Following Blau and O'Loughlin, the models of CSV matrix big-bangs are generalised from flat space to singular homogeneous plane waves, setting the stage for a discussion of fuzzy sphere behaviour in regimes of strong and weak coupling. A fair amount of background material has been included in this work to strike the balance between an ample introductory text and a rather terse research paper. The advanced reader might want to skip these parts and go straight to the relevant sections. Notably, this work includes the unabridged reprints of two publications : M. Blau, D. Frank and S.Weiss, "Fermi coordinates and Penrose limits", Class. Quant. Grav. 23 (2006) 3993-4010, hep-th/0603109. in section 1.6, with a further (unpublished) example of the new techniques in 1.7, and, M. Blau, D. Frank and S.Weiss, "Scalar field probes of power-law space-time singularities", JHEP 08 (2006) 011, hep-th/0602207. reprinted in section 2.4. Apart from those publications, chapters three and four contain original work not (yet) published in - section 3.2, mostly a derivation of the membrane matrix model from a perspective complimentary to the usual one found in the literature, - section 3.5, where we employ the $R_\xi$-gauges not used before on the BMN matrix model to detail the physical relevance of the one-loop effective potential, - section 4.3, most prominently the scaling behaviour of the matrix models and - section 4.4 on fuzzy sphere dynamics in matrix big-bangs. As the title tells, plane waves and light-cone gauge, (power-law) space-time singularities and fuzzy spheres in matrix models are the main threads running through this work. In its diversity, the present report surely does reflect an important quality of string theory. After many surprises and indeed revolutions the theory has turned into a broad frame-work that allows for the exploration of a wealth of new ideas and theoretical phenomena to sharpen our tools and senses for the experimental results soon to come.